Nash Ranges for the Push-or-Fold Play

Introduction

Do these adaptations depend on the size of the ICM effect (bubble factor)?

Good heads-up play is vital if you want to play SNGs successfully. As the blinds are usually very high by the time you reach the HU, you will mainly find yourself in situations in which you fold your hands preflop or push them all in - this is known as push-or-fold play.

So-called nash ranges (NRs) play an important role in this context. Since you will often be up against a fish in the heads-up, who has built his stack through crazy moves, you need to be cautious when using such nash ranges for your game.

It is not necessary to have read them in order to understand this article. The so-called SAGE is basically based on Nash equilibria; SAGE simplifies them in a way that they are somewhat easier to memorise. However, this also makes the system rather stiff.

Basics

Abbreviations

BB

Big blind

NCR

Nash calling range

BL

Blind level

NPR

Nash pushing range

CR

Calling range

NR

Nash ranges

ES

Effective stack

P/F

Push-or-Fold

EV

Expected value

PP

Pocket pair

HEV

Hand EV

PR

Pushing range

HU

Heads-up

REV

Range EV

ICM

Independent Chip Model

SB

Small blind

First of all we need to clarify some basic terminology:

Effective stack

The effective stack (ES) is an important parameter in the analysis of push-or-fold situations in heads-up. It refers to the size of the smaller of the two stacks before the blinds (and antes) are posted, measured in BBs. Therefore, the ES is always the same for both players.

EXAMPLE 1:

The blind level (BL) is 500/1000; after you pay the SB, you are left with 5500 chips, your opponent in the BB has 8000 chips left. In this situation, ES = 6.

Ranges

A range in poker refers to a certain selection of hands. When it comes to push-or-fold situations, particularly the "x % ranges" are relevant: They consist of the strongest x % of the 1326 possible starting hands.

However, this definition is not clear-cut since the relative strength of a hand always depends on the specific situation, and this is also true for push-or-fold situations.

EXAMPLE 2:

If ES = 9, the calling range (CR) of 42.7% comprises the hands 22+, Ax+, K2s+, K4o+, Q4s+, Q8o+, J7s+, J9o+, T8s+, T9o, 98s. The + means that all hands with a better kicker (or bigger PPs) are also included in the range. A hand such as K4 is therefore part of the range, while 78s is not.

Compare this to the 43.3% pushing range (PR) for ES = 16. It consists of the hands 22+, Ax+, K2s+, K7o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T9o, 96s+, 98o, 85s+, 75s+, 65s, 54s. You see that K4, K5 and K6 are not included here, while 54s is.

The example illustrates that a range of 43% for a call with ES = 9 does not equal the range for a push with ES = 16. That's why we will always list the relevant hands in the following illustrations. The %-figures and the precise ranges are taken from the ICM calculator at www.holdemresources.net.

It's easy to understand that the following applies as well:

If you add a random hand to a % range, you will get a new % range when that hand is the strongest hand outside of the range. Vice versa, if you remove a hand from a % range, you will get a new % range when that hand is the weakest hand within the range.

ICM

You have already encountered the Independent Chip Model in some previous articles:

A HU is characterised by the fact that ICM effects are no longer in place. That's why the HU in an SNG (as well as the HU in an MTT) is basically the same as the HU in a cash game, in which every chip is worth

[($ 1st place) - ($ 2nd place)] / (Total number of chips).

Whether you should push your hand or call a push in a HU therefore mainly depends on the following parameters:

Your ES

Your opponent's calling or pushing range

Aside from the HU, there is always a certain level of the ICM effect in MTTSs and SNGs. In SNGs, this effect tends to have its peak on the bubble and is therefore also known and measured as the bubble factor (see The Bubble Factor in different SnG formats).

That's not the entire article...

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Comments (3)

Nice explanations, especially how +REV could include some -HEV hands.<br />
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2 questions: (1) how come Nash ranges table has some minor differences compared to Nash HU charts from www.holdemresources.net (e. g you can't push 85o with ES 4BB according to latter, but you can according to the former). (2) In the chapter Modification guidelines, I'm not sure to what sentence "cases 2 and 4" in the brackets refers to - against most opponents in the lower limit STT, i think i have to apply rules 2 and 3. <br />
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Thanks for the article.